Theorem cbvoprab3v

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004) (Revised by David Abernethy, 19-Jun-2012)

	Ref	Expression
Hypothesis	cbvoprab3v.1	$$⊢z=w\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvoprab3v	$$⊢\left\{⟨⟨x,y⟩,z⟩|\phi \right\}=\left\{⟨⟨x,y⟩,w⟩|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvoprab3v.1	$$⊢z=w\to \left(\phi \leftrightarrow \psi \right)$$
2		nfv	$$⊢Ⅎw\phi$$
3		nfv	$$⊢Ⅎz\psi$$
4	2 3 1	cbvoprab3	$$⊢\left\{⟨⟨x,y⟩,z⟩|\phi \right\}=\left\{⟨⟨x,y⟩,w⟩|\psi \right\}$$